**Year** |
**1790** |
**1800** |
**1810** |
**1820** |
**1830** |
**1840** |
**1850** |
**1860** |

**Population****(in thousands)** |
**3929** |
**5308** |
**7240** |
**9639** |
**12861** |
**17063** |
**23192** |
**31443** |

#### Answer

P(t)=(4.503×10−20)(1.0299)t

#### Explanation

The exponential function to the given data using graphic calculator is

P=abt

Taking natural logarithm on both sides

lnP=lnabt=lna+lnbtlnP=lna+tlnbY=A+BtWhere Y=lnQ,A=lna,B=lnb

To find the values of a,b the Normal Equations are

∑Y=nA+B∑t∑tY=A∑t+B∑t2

P(t)=abt where a=4.502744×10−20 and b=1.029953851

Since P(t)=abt, we can find the first derivative

P′(t)=abtlnb

We have P′(1800)≈157 and P′(1850)≈686

**Conclusion: **

P(t)=(4.503×10−20)(1.0299)t

#### To determine

**b)**

**To estimate:**

the rates of population growth in the year 1800 and 1850 by averaging slopes of secant lines

#### Answer

P′(1850)=m1+m22=719

#### Explanation

For the year 1800, slopes of secant lines are given by

m1=5308−39291800−1790=137.9m2=7240−53081810−1800=193.2

So by averaging of the secant lines, we get, P′(1800)=m1+m22≈166 thousand people/year

For the year 1850,

m1=23192−170631850−1840=612.9m2=31443−231921860−1850=825.1

So by averaging of the secant lines we get, P′(1850)=m1+m22=719 thousand people/year.

**Conclusion:**

The rates of population growth in the year 1800 and 1850 by averaging slopes of secant lines is 719.

#### To determine

**c)**

**To estimate:**

the rate of growth in the year 1800 and 1850 in part (a) and compare the results with part(b)

#### Answer

The values using averaging of the secant lines are much higher than the values using the exponential curve

#### Explanation

From part (a) P′(1800)≈157 and P′(1850)≈686

From part (b) we have P′(1800)≈166 and P′(1850)≈719

**Conclusion:**

The values using averaging of the secant lines are much higher than the values using the exponential curve.

#### To determine

**d)**

**To predict:**

The population in the year 1870 using the exponential model and compare it with the actual population 38,558,000.Explain the discrepancy

#### Answer

P(1870)≈41946763.The difference of population of approximately 3.4 million is attributed to the civil war in the year 1861-1865.

#### Explanation

The exponential function to the given data using graphic calculator is

P(t)=abt Where a=4.502744×10−20 and b=1.029953851

P(1870)=41946.76319thousand people per year≈41946763

The difference in the population is 41946763-38,558,000 = 3388763

The reason for the difference of population of approximately 3388763 people is attributed to the civil war in the year 1861-1865.

**Conclusion:**

P(1870)≈41946763.The difference of population of approximately 3.4 million is attributed to the civil war in the year 1861-1865.